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Periods of modular forms and Jacobi theta functions

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1991). Periods of modular forms and Jacobi theta functions. Inventiones Mathematicae, 104(3), 449-465. doi:10.1007/BF01245085.


Cite as: http://hdl.handle.net/21.11116/0000-0004-3923-5
Abstract
In an earlier paper \\it W. Kohnen and \\it D. Zagier [Modular forms, Symp. Durham 1983, 197-249 (1983; Zbl 0618.10019)] introduced the period polynomial r\\sb f(X)=\\int\\sb 0\\spi∞f(τ)(τ-X)\\spk-2dτ for a cusp form f of weight k in the context of the Eichler-Shimura isomorphism. There they also derived a formula for the (rational) coefficients of a related polynomial in two variables.\\par In the paper under review the author gives a more attractive formula by introducing a generating function. First of all the definition of r\\sb f(X) is extended to f\\in M\\sb k, the space of elliptic modular forms of weight k. Then the generating function is \\align C(X,Y;τ,T) amp; = (XY-1)(X+Y)\\over X\\sp 2Y\\sp 2T\\sp-2 \\\\ amp; +\\sum\\sp ∞\\sbk=2\\sum\\sbf\\in M\\sb k\\atop\\texteigenformr\\sb f(X)r\\sb f(Y)-r\\sb f(-X)r\\sb f(-Y)\\over 2(2i)\\spk-3(f,f)(k-2)! f(τ)T\\spk- 2,\\endalign where (f,f) is the Petersson scalar product. If \\Theta(u)=\\Theta\\sb τ(u) denotes the Jacobi theta function, one obtains the surprising identity C(X,Y;τ,T)=\\Theta'(0)\\sp 2\\Theta((XY-1)T) \\Theta((X+Y)T)\\over \\Theta(XYT) \\Theta(XT) \\Theta(YT) \\Theta(T). The right hand side can also be rewritten, where the Eisenstein series G\\sb k, k≥ 2, are involved in place of the theta function.